Inverse function / mapping considering vector multiplication by matrix

Question

Inverse function / mapping considering vector multiplication by matrix also touches symetric encryption

Consider, there's a simple matrix as a mapping from R3 ➝ R3

R3 ➝ R3: v(3) x M(3,3) = v(3)

⎡ 1 ⎤   ⎡ 4 0 0 ⎤   ⎡ 4 ⎤ 
⎢ 2 ⎥ x ⎢ 0 3 0 ⎥ = ⎢ 6 ⎥  
⎣ 3 ⎦   ⎣ 0 0 2 ⎦   ⎣ 6 ⎦ 

Then it ∃ also an inverse function as a simple mapping matrix from R3 ← R3

v(3) x M(3,3) = v(3) R3 ← R3
                          
⎡ 4 ⎤    ⎡ ¼ 0 0 ⎤     ⎡ 1 ⎤    
⎢ 6 ⎥  x ⎢ 0 ½ 0 ⎥  =  ⎢ 2 ⎥ 
⎣ 6 ⎦    ⎣ 0 0 ⅓ ⎦     ⎣ 3 ⎦   

Suppose, there is a trivial matrix(3x2) as a mapping from R2 ➝ R3

R2 ➝ R3: v(2) * M(3x2) = v(3)

    ⎡ 1 ⎤   ⎡ 4 0 ⎤    ⎡ 4  ⎤
    ⎣ 2 ⎦ x ⎢ 0 3 ⎥  = ⎢ 6  ⎥ 
 ⎢⎢         ⎣ a b ⎦    ⎣ ab ⎦  

Then also there ∃ a matrix(2x3) an inverse function as reverse mapping from R2 ← R3

R3 ➝ R2: v(3) * M(2x3) = v(2)

⎡ 4  ⎤   ⎡ ¼ 0 0 ⎤   ⎡ 1 ⎤ 
⎢ 6  ⎥ x ⎣ 0 ½ 0 ⎦ = ⎣ 2 ⎦
⎣ ab ⎦

Question for all mathematicians:

For which non-trivial matrices does also ∃ an inverse function to / reverse mapping?

Simple Matrices are

a 0 0 0
0 b 0 0
0 0 c 0
0 0 0 d
or
0 0 a 0
0 b 0 0
0 0 0 c
d 0 0 0

Non simple matrices are if there is more then one variable per row or column.

Kind regards,

Heinrich Elsigan.

post scriptum: I know, that when mapping higher to lower dimension, that there's certainly not any inverse mapping, because of loss of information / complexity.

My question is, how big is the numnber of not simple matrices having an inverse mapping (needed for symetric encryption school contest)